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The distribution of dark matter around galactic or cluster halos has usually been assumed to be approximately isothermal with a non-zero core radius, which is expected to be of the order of the size of the visible matter distribution. Recently, the possibility has been raised that dark matter halos might be singular in the sense that the dark matter density could increase monotonically with radius r down to a very small distance from the center of galaxies or clusters. Such central cusps in the dark matter density could lead to a high ux of gamma rays from WIMP dark matter annihilation. Here we analyze two possibilities that have been discussed in the literature, / r n with n 1 or 2, and point out that such density pro les are excluded by gravitional lensing analyses on cluster scales and by the rotation curves of gas-rich, halo-dominated dwarf spirals on small scales. We also point out that if spiral galaxies form by gas infall inside dark matter halos, as they are expected to do in any hierarchical clustering model, such pro les almost always lead to falling rotation curves after infall, contrary to observations.

Using six high‐resolution dissipationless simulations with a varying box size in a flat Lambda cold dark matter (ΛCDM) universe, we study the mass and redshift dependence of dark matter halo shapes for Mvir= 9.0 × 1011− 2.0 × 1014 h−1 M⊙, over the redshift range z= 0–3, and for two values of σ8= 0.75 and 0.9. Remarkably, we find that the redshift, mass and σ8 dependence of the mean smallest‐to‐largest axis ratio of haloes is well described by the simple power‐law relation 〈s〉= (0.54 ± 0.02)(Mvir/M*)−0.050±0.003, where s is measured at 0.3Rvir, and the z and σ8 dependences are governed by the characteristic non‐linear mass, M*=M*(z, σ8). We find that the scatter about the mean s is well described by a Gaussian with σ∼ 0.1, for all masses and redshifts. We compare our results to a variety of previous works on halo shapes and find that reported differences between studies are primarily explained by differences in their methodologies. We address the evolutionary aspects of individual halo shapes by following the shapes of the haloes through ∼100 snapshots in time. We determine the formation scalefactor ac as defined by Wechsler et al. and find that it can be related to the halo shape at z= 0 and its evolution over time.

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